Arithmetical completeness for some extensions of the pure logic of necessitation

Abstract

We investigate the arithmetical completeness theorems of some extensions of Fitting, Marek, and Truszczyński's pure logic of necessitation $\mathbf{N}$. For $m,n\in \omega$, let ${\mathbf{NA}}_{m,n}$, which was introduced by Kurahashi and Sato, be the logic obtained from $\mathbf{N}$ by adding the axiom scheme ${\square }^{n}A\to {\square }^{m}A$. In this paper, among other things, we prove that for each $m,n\ge 1$, the logic ${\mathbf{NA}}_{m,n}$ becomes a provability logic, that is, there exists a provability predicate ${\mathrm{Pr}}_T(x)$ of $T$ whose $T$-verifiable modal principles are exactly the logic ${\mathbf{NA}}_{m,n}$.

29 pages